Grosjean played out of his skin on the indoor clay court, showing every shot in the book in a superbly athletic display which left Kafelnikov a broken figure on the baseline.
Earlier, Russia’s Marat Safin drew on his greater experience of the big occasion to beat Davis Cup debutant Paul-Henri Mathieu 6-4 3-6 6-1 6-4 in the opening singles.
A netted forehand gave Safin his first break point at 4-3 which Mathieu saved by wrong-footing his opponent who ended up sprawled in the clay at the baseline.
But the Russian champion earned a second break point immediately and a wild forehand by the Frenchman gave away the game and a 4-3 lead.
A Mathieu forehand gave Safin his first set point which he wasted but his fifth ace of the match earned him the set.
A forehand gave him a break point in the second game and a backhand volley won him the game.
Mathieu had three more breakpoints for a 4-0 lead as Safin looked to lose confidence in his own shot-making. The Russian saved all three but Mathieu, spurred on by 14,000 spectators, held his own serve to take a 4-1 lead.
A forehand cross-court from the baseline by Mathieu followed by a Safin double fault earned France two set points in the eighth game, and a third after the Russian netted a simple forehand.
Safin saved them but could not prevent the Cup debutant wrapping up the set a game later when Safin netted a backhand.
Safin broke Mathieu in the second game as the Frenchman tried one too many ambitious shots and won nine out of 10 points to take a 5-0 lead and went on to clinch the third set 6-1.
Mathieu hit trouble again in the third game of the fourth set and allowed Safin four break points which he eventually took advantage of to take a 2-1 lead.
Safin broke again to serve for the match at 5-2. Astonishingly he wasted his first match point with a double fault and two superb passes by Mathieu won him the game.
RESULTS
Sebastien Grosjean (Fra) bt Yevgeny Kafelnikov (Rus) 7-6 (7/3), 6-3, 6-0; Marat Safin (Rus) bt Paul Henri Mathieu (Fra) 6-4, 3-6, 6-1, 6-4
Saturday (from 1300 GMT)
Doubles: Nicolas Escude/Fabrice Santoro (Fra) v Yevgeny Kafelnikov/Marat Safin (Rus)—Reuters